A NNALI
DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA
Classe di Scienze
B JÖRN G USTAFSSON
M IHAI P UTINAR
An exponential transform and regularity of free
boundaries in two dimensions
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 26, no 3
(1998), p. 507-543.
<http://www.numdam.org/item?id=ASNSP_1998_4_26_3_507_0>
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Ann. Scuola Norm. Sup. Pisa Cl. Sci.
Vol. XXVI (98), pp. 507-543
(4)
An Exponential Transform and Regularity of
Free Boundaries in Two Dimensions
BJÖRN GUSTAFSSON -
MIHAI PUTINAR
To professor Harold S. Shapiro on the occasion of his seventieth birthday and to
the twenty-fifth anniversary of his quadrature domains.
Abstract. We
investigate
exp 7r 0
E(z, w) =
compute it in
the basic
x
properties
dA(~)
-]) (z,
The main result
w E
of the
C) of
transform
domain Q c C and
exponential
a
states that if the Cauchy transform of the characteristic function of Q has an analytic continuation from C B S2
across aS2 then the same is true for E(z, w), in both variables. If F (z, w) denotes this analytic-antianalytic continuation it follows that 8 Q is contained in a
real analytic set, namely the zero set of F (z, z). This gives a new approach to the
regularity theory for free boundaries in two dimensions.
some
simple cases.
Mathematics Subject Classification (1991): 32D15
47B20 (secondary).
(primary), 30E20, 35R35,
1. - Introduction
The exponential of the Cauchy transform of a function of a real variable
appears in the canonical representation, of several classes of analytic functions in
the upper half-plane. One of these representations was used by R. Nevanlinna
in his study of the moment problem on the line; later the same exponential expression was related to perturbation determinants of selfadjoint operators,
see [K-N, Appendix]. The present paper is devoted to the analytic aspects of
a two-dimensional analogue of exponentials of Cauchy transforms which have
appeared in operator theory as a generalization of perturbation determinants.
A specific instance of such a generalization is a formula, due to R. W. Carey
and J. D. Pincus, for the determinant of the multiplicative commutator resolvent
for a pure hyponormal operator T with rank one self-commutator. It reads as
Pervenuto alla Redazione il 24 febbraio 1997
e
in forma definitiva il 6 ottobre 1997.
508
follows.
In this formula the function g satisfies 0
g 1 and is a kind of "spectral
T and it characterizes T up to
of
the
parameter", called
principal function,
The
member
of
(1.1) is what we call the exponential
unitary equivalence.
right
transform of g.
Quite independent of its operator theoretic importance the exponential transform turns out to be both interesting and useful from a pure function theoretic
standpoint. This is basically the point of view we take in the present paper.
First, in Sections 2 and 3, we derive some basic properties of the exponential
transform and give a number of examples, mainly from the theory of quadrature
domains.
Our main results then appear in Sections 4 and 5, and they concern analytic
continuation properties of the exponential transform and applications of this to
regularity questions for free boundaries. In fact, we prove that if the Cauchy
of a domain Q has an analytic continuation from outside S2
transform
across
then the same will be true, in both variables, for the exponential
transform
w) of g XQ.
If F(z, w) denotes this analytic continuation (F will be analytic in z,
0 in Q and on (most
antianalytic in w) it follows, using that
that
of)
=
Thus a S2 is contained in a real analytic set. This is the kind of regularity
statement we obtain for boundaries a S2 admitting analytic continuation of the
Cauchy transform. Then one can easily go further to obtain complete classification of possible singular points on a 0, but this we do not discuss because
all this has already been done.
Indeed, there is a complete regularity theory for free boundaries in two dimensions of the type we are considering here, due to M. Sakai [Sa3], [Sa4],
[Sa5]. Sakai even starts from weaker assumptions than we do, so our regularity
result is just a special case of Sakai’s theory. Nevertheless we think there are
some good points with our approach: it is new and completely different from
Sakai’s approach, it is both technically and conceptually simple and it directly
provides a good defining function for the free boundary.
Our proof uses a few real variable tools, namely a sequence of exhaustion
functions due to L. Ahlfors and L. Bers and a lemma, due to L. Caffarelli,
must have area
L. Karp, A. Margulis and H. Shahgholian, showing that
measure zero. The rest of the proof consists of complex analysis, and for this
we actually give three different approaches, two function theoretic ones and
one operator theoretic, the latter in Section 5. In Section 5 we also obtain a
509
somewhat stronger version of the
resolvent extends.
analytic continuation, saying
that
a
certain
ACKNOWLEDGEMENTS. Most of this paper grew out during a couple of
weeks at KTH, and the material was then daily discussed with Peter
Ebenfelt, Lavi Karp, Henrik Shahgholian, Harold S. Shapiro and occasionally
others. We thank them all for their generous contribution of ideas and for letting
us include some of their unpublished results here (see in particular Lemma 4.4).
We also thank Siv Sandvik for typing our manuscript. The second author thanks
the Department of Mathematics of KTH for its hospitality.
This work has been supported by NFR (Sweden) grants Nos. M-AA/MA
08793-310 and R-RA 08793-313 and NSF (USA) grant No. 9500954.
summer
SOME
NOTATIONS USED
dA
area measure
2. - Definition and
elementary properties
DEFINITION 2.1.
defined as
In
case
g
=
(two dimensional Lebesgue measure) in C.
For
0
g
XQ for some C C
E
we
the
exponential
write Egz in
place
of
transform
Exo.
Thus
of g
is
510
The above definition
requires
some
justifications. First, if z
=
w
then
where the integral is either a finite number &#x3E; 0 or +00. In the latter case
Eg (z, z) is understood to be zero.
If
then we are integrating a locally integrable function and problems
can only arise for large ~ .
However,
is
integrable
over
all C and
for large [§I (see more precisely Lemma 2.4 below). Thus the negative part of
Re
is integrable so that f, Re
either is a finite real
§-z §-w&#x3E;
(-z)(-w)&#x3E;
In the latter case Eg(z, w) is defined to be zero, otherwise it
number or
is a nonzero complex number.
In summary, Eg (z, w) is a well-defined complex number for any pair (z, w) E
g()-
-g-w dA()
(c2.
REMARK 2.2. It is easy to
then
see
that the
otherwise Eg(z,
if supp g is not compact.
whenever
We recall here also the definition of the
(say) compact support it is
Cauchy
Eg(z, w) - 0,
occur
with
following
only
holds: if
Clearly (2.2)
can
transform:
and satisfies
in the sense of distributions. More generally, the Cauchy transform may be
defined for any distribution p of compact support as the convolution between It
of aZ.
and the fundamental
7rz
In the rest of this section we go through some of the basic properties and
a couple of examples of the exponential transform, usually for the case g = XQ,
Q c C. Much of this material is not new (cf. in particular [MP], [PI], [P3]) and
some is very elementary (like the first property below). For a characterization of
which functions of two complex variables are of the form Eg(z, w) for some g,
solution --L
see
[P3].
511
a) SCALING
PROPERTIES
For 0 g, gj E
L°°(C),
a &#x3E;
0
EXAMPLE 2.3. We compute E(z,
computing the integral
in the four components of
For Z, W
E
D’
Hence
in this case. _
For Z E Uc,
Thus
W
e D
(C B
w) for the unit disc D and
z,
w / 8D by
512
The
case z E
For Z, W
D, w
D,
E
Thus
In summary:
E
Ec
is obtained from the
finally
previous
one
by (2.3).
513
b)
BEHAVIOUR
UNDER
MBBIUS
TRANSFORMATIONS
Under Mbbius transformations
E~ behaves
(ad -
If
as
follows. If
c
=
0, then
then
(of interest only if -djc f/ Q) and
Here C &#x3E; 0 is
a
constant
depending
on S2 and
f.
If
-d /c
E Q then C is
strictly positive.
PROOF. We assume that
the change of variable
Making
identity
one
~
=
0 since otherwise all statements are trivial.
in the integral in (2.1 ) and using the
obtains
This
equals Egz(z, w)
if
c
=
0
(proving (2.5))
and
equals
if
at least if all the integrals appearing are absolutely convergent, i.e. if
all
occuring are nonzero. If some E~ vanishes it is easy to see that (2.6)
still holds.
For -d /c E S2, i.e. for
containing a neighbourhood of infinity, equation (2.6) is not of much use because both members then are identically zero.
514
However, applying (2.6)
multiplying
to
both members
Q B
by
r), where
r) C Q, and then
w) gives
Since
we
find
Let
now r ~
0. It is
easily
verified then that, for
z,
w=,4 - d /c,
for suitable constants C1, C2 &#x3E; 0. Putting the pieces together we obtain the
desired formula (2.7) in the case that -d/c E Q.
n 0 since
For -d/c ¢ S2 (2.7) holds with C = 0
contains
a neighbourhood of infinity). If -d/c E aS2 then f(Q)C still is unbounded
* 0, so that (2.7) holds with C = 0. In the
and typically
exceptional
cases that
small modifications of the proof show
0 (for -d /c E
that (2.7) still holds (with C &#x3E; 0).
D
C) BEHAVIOUR
AT INFINITY
has compact support so that Eg is analyticWe assume that 0 g E
antianalytic in a neighbourhood of infinity. Expanding the integral in the definition of Eg(z, w) in power series in 1 /z and/or 1 /w gives
515
1 w-
as
oo;
00, with z e C fixed.
the
Thus
Cauchy transform is
of the exponential transform.
asIwl
-~
a
derivative at
infinity
in
one
of the variables
d) ESTIMATES
We
only treat
nonpositive, hence
If
w e
product
the
case
g =
If
,z = w
then the exponent in
(2.1 ) is
be the disc with diameterz - wi and z,
Since Re(~ - z)(~ - w) has the interpretation of being the scalar
between § - .z and § - w considered as vectors we have
let
Thus
with equality if and only if SZ = lDz,w up to null-sets. By (2.5)
an absolute constant, in fact EDZ, w (.z, w) = 2 by Example 2.3.
More specifically we may decompose
’
(z, w) is
516
The integrands in the Ej are:
for
0 and integrable,
for E2 : &#x3E; 0 but possibly nonintegrable,
for E3: integrable. Thus
-
-
In summary:
LEMMA 2.4. For any Q C (C, z, w
E
C
with equality if and only if Q is a disc (up to null-sets) and z, w diametrically
opposite points on the boundary. Furthermore, for fixed z, WEe the set function
and decreasing as a function
S2 f--~ ~ (z, w)is increasing as
Finally,
of Q B
For z E Q, let r = dist (z, Q’). If W E D(z,
Hence the second part of Lemma 2.4 shows that
r) then Dz,w
C
D(z, r)
c Q.
where we used also Example 2.3 and (2.5). If w ¢ D(z, r) then the right
member in (2.8) is &#x3E; 1. Therefore we have in any case, using also Lemma 2.4,
for Z
E
S2.
By symmetry
we
LEMMA 2.5. For any Z,
e) CONTINUITY
The
get:
W E
Q,
PROPERTIES
following example
shows that
E (z, w) need
not be continuous on CxC.
517
EXAMPLE 2.6. For
Thus
S2a is
point
if 0
Let E
a &#x3E;
0,
set
domain in the
a
1,
a
a
right half-plane with 0 E
Lipschitz comer if a 1 and
=
a
an
outward cusp if
a &#x3E;
1.
continuous
at
is smooth,
or
We compute
=
which is = 0 if 0 a 1 but &#x3E; 0 if a &#x3E; 1.
Since E(z, z) = 0 for all z E S2a it follows that
(0,0) if a &#x3E; 1.
Now for
If
even
C~1 smooth boundary
general
a
0 this
domain S2 c C set
means
Clearly Z is an exceptional subset of
Lipschitz. Some simple observations are
(i) Q has vanishing Lebesgue density at
z E
-
Z is
an
in
each
point
of
E
z E
0
always
dense in
z ; then any closest
and n
(iii) If
It is empty if
Z then 0
(ii) a S2 B
z,
E is not
(Proof:
neighbours ~n of
if
z E
z, on 8Q
(Proof: if
0.)
take zn E Q,
Z.
as E --~
satisfy §n
-~
zn
Z, the latter in view of (i) and the fact that a Q satisfies
inner ball condition at ~n . )
0 then Egz is always discontinuous at each
point
of Az (as
Example 2.6).
PROPOSITION 2.7. For any open set Q C C, EQ (z, w) is continuous on C2 B 0 z.
On Az, EQ is still continuous in each variable separately when the other variable
is kept fixed.
518
NOTE: Thus Egz is continuous in all C
case
x
C if and
only
if Egz w 0
or
Z
=
0.
PROOF. Since E~ clearly is continuous if E == 0 we may exclude this
off the diagonal. When
from discussion. Then, by Remark 2.2,
continuity of E~ at (z, w) is the same thing as continuity at (z, w)
which is immediately verified when
and also
(z, w)
dA~ ~~ w) ,
of I
when z = w E
For the first statement of the
remains to consider points (z, z) with z E
at which
z ) 0.
Let (zn , wn ) -~ (z, z) and let Dn
of Lemma 2.4)
=
proposition it therefore only
i.e. exactly those points
=
is
a
to
XQ(~)II~ _ Z12 .
sequence of
nonnegative
Therefore
Then
(cf. the proof
functions
converging pointwise almost everywhere
by assumption and Fatou’s lemma
The integrals in the right member of (2.12), which hence tend to +oo, are
The remaining parts of the
parts of the integrals in the exponents of E (zn ,
bounded
contributions
to
E (Zn, wn) as shown in the
exponents give uniformly
the
of
of
Lemma
2.4.
notation
that
2,
(In
proof
proof E2 - 0, while 1
=
=
0
the
first
Thus
the
of
1.)
E (zn , wn ) -~
E (z, z) completing
IE31I
proof
statement of the proposition.
The separate continuity on Az is a previously known fact [Cll ] [MP, p. 258]
but let us still give a short proof. We need to prove that E (zn, z) - E(z, z)
as zn -~ z for z E Z. This is equivalent to proving that
- z, where by assumption the right member is finite.
For notational
Q
convenience we assume that z
C
0,
D(0, 1).
1. Since [§[ &#x3E;
So let zn - 0 and let 0 E
for ~
we have
as zn
=
’
519
Here the first term equals 27r 8/( 1 - ~)2, hence can be made arbitrarily small
by taking 8 small enough, and for any fixed E the second term tends to zero
as z, - 0 by Lebesgue’s dominated convergence theorem.
This finishes the proof of Proposition 2.7.
0
f)
DERIVATIVES
AND GENERAL STRUCTURE OF
E(z, w)
Since E is a bounded function it is also a distribution in (C2. We shall
some of the classical and distributional derivatives of E =
c C
open. We exclude the case E - 0 from discussion.
0 for w E ac, i.e. E(z, w)
It is clear that az E
0 for z E S2~ and aw E
is analytic in z, antianalytic in w when the variable in question is outside SZ.
One trick one may use for computing, e.g., derivatives 8z inside S2 is to excise
a small disc D C S2 containing z and write (by (2.4))
compute
=
=
Since the second factor is analytic in z, 9~ will only act
which we have explicit expressions by Example 2.3.
If D
D(a, r) and also w is in D this gives
=
The
same
Thus
expression is obtained for w V D.
(using also (2.3))
Handling higher
derivatives in the
same
way
gives
on
the first factor, for
520
It also follows that
ponents of (C B
as
EQ(z, w) has the
E~ has in (C B
where F, G, H are functions which
with (2.13), (2.14) gives that
Since E (z, z)
follows that
&#x3E;
0 for z
E
QC
are
we
same
type of
structure in the com-
namely
analytic
in their arguments.
have F (z, z)
&#x3E;
0 for
z E
S2c,
Comparison
and it also
for
Indeed, for z, w E D
c
Q,
(D
a
disc
(2.18) follows from the corresponding
(2.15) and by letting w - z.
so
as
above)
we
have
statement for the disc D
along
with
Next we take contributions across aQ into account by computing derivatives
the
distributional sense in all C2 . For computing the distributional derivative
in
we
are allowed to use the chain rule ([E-G, Section 4.22]) and the fact
azE
that the exponent in E(z, w) as a function of z is the Cauchy-transform of the
function
in all
«
(~- - w- ) -’ xgz(§), which is in
(C) for all
p
2. This
gives
(C2. Similarly,
Note that the right members are locally integrable functions. Thus there
distributional contributions on aQ for the first order derivatives.
are no
521
For the
we
computation of j2 E we
test function q;.E D
have, for any
that aS2 is smooth. In this
and using (2.19), (2.15).
assume
(((:2)
case
a distribution f on a test function cp.
Let n denote the outward unit normal vector on aQ considered as a complex
inds where d s is arc-length measure on
Then d z
valued function on
Here(f, cp) denotes the action f(cp) of
=
We conclude that
Note that
a2E
equals
the
complex
Q9
dA).
computed similarly.
For
z- w
is
-2
n(z)
where we used E(z, z) = 0, z E Q,
estimate (2.9). Thus at least in the
measure
azawe
we
of E at these points and the
of iterated integration, as written
continuity
sense
have
522
above,
we
have
We do not know whether the right member here always is a locally integrable
function.
One way to get a statement which only involves absolutely integrable functions is to replace, in the above computation, occurrences of factors
(or
integrations over S2 with respect to z) by functions 1/1 n (z) which are compactly
X~ pointwise as n - oo. This
supported in S2 and satisfy 0
that
gives the statement
in the
of distributions as n -~
because of Lemma 2.5.
sense
Lloc (C2 )
3. - Further
oo.
Here the left members
certainly
are
in
examples
a) CLASSICAL QUADRATURE DOMAINS
A (classical, bounded) quadrature domain is
the property that there exists
of points in Q such that
a
a bounded domain Q C C with
distribution A with support in a finite number
for every integrable analytic function w in Q.
The basic example is any disc Q
1I))(a, r), in which case
In
the
of
on
action
/t
cp analytic is of the form
Jrr2cp(a). general
-
for suitable
n =
... , zm C
C, nk &#x3E; 1, Ckj
E
C. Set then
(/1, cp)
=
(assuming
nk and
It is known
curve, more
[Ah-Sh], [Gul], [Sh2] that, with Q, J1
precisely
that
as
above, aQ is
an
algebraic
523
where
Q(z, w) is
an
irreducible
polynomial
of the form
with
and ann = 1. For S2 = D(0, 1) we have P(z) = z, Q(z, w ) =
1 for example.
The "finite set" in (3.2) is a subset of the set of isolated points in f z E C :
Q (z, z) = 0) and this subset can be chosen arbitrarily.
Clearly Q determines it, P and Q as above uniquely when Q is a quadrature domain
is determined only as an analytic functional). Conversely, Q
determines Q up to the finite set in (3.2).
for z E QC in (3.1 ) gives that
Taking w(§)
zw -
_~1Z
and
by an approximation argument (using [Be]) this equation is in fact found
equivalent to S2 being a quadrature domain for /t [Ah-Sh], [Sal], [Sh2].
Note that /1 here is a rational function vanishing at infinity. Since any such
rational function is the Cauchy transform of some distribution it as above it
follows that S2 is a quadrature domain if and only if X~ coincides on QC with
to be
a
rational function.
It is convenient to set
for
(3.3) holds. Then S(z) is meromorphic in Q and satisfies
when
z
thus S(z) is the
it follows from
Schwarz, f’unction for aQ [D], [Sh2].
(3.4) and analyticity of S(z) that
Since
Q (z, z)
0.
Hence S(z) is the algebraic function defined by Q(z, w)
The exponential transform for a quadrature domain with data
computed in [PI] to be
=
0
on
a S2
=
as
above
was
E
Moreover, using operator theory a strong converse of this result
obtained, namely saying that whenever an exponential transform Eg(z, w)
with 0 g 1 is of the form of the right member in (3.6) for large
was
then g
=
XS2
a.e.
for
a
quadrature
domain Q
as
above.
524
Schwarz function at hand it is easy, in view of continuity
and general structure (2.15) results, to compute E everywhere
within the realm of S (z) once it is known in S2~ x ff. When, for example,
should by (2.15) acquire a factor
z is moved from Qc to Q then
z - M). In order for the transition to be continuous one has to compensate
in the denominator. Thus E(z, w) acquires the factor
by a factor
(2 - W-) / (S(z) compared to its natural analytic continuation given (in the
case of quadrature domains) by the right member of (3.6). (These matters will
be discussed in more depth in Section 4.)
In this way we obtain the following formulas, complementing (3.6).
Having the
(Proposition 2.7)
EXAMPLE 3.1. For the functional
where r &#x3E; 1, there turns out to be
data P and Q are
an
essentially unique Q,
and the
corresponding
(see [Sh2]). Thus
for z, w E nc. For z, w in the remaining parts of C,
from (3.7)-(3.9) inserting the Schwarz function (determined
The "finite set" in
(3.2) either is empty
case.
EXAMPLE 3.2. For
or
consists of just
z
w) is obtained
by (3.5), (3.10))
=
0 in the present
525
there also is
a
unique quadrature
domain Q [Ah-Sh]. This is
given by
the data
Thus
--c
for z, w E Q .
We remark that
is a cardioid and that it has an inward cusp at z
0.
(Since Q (z, z) vanishes to the second order at z = 0, like in the previous
example, this point is a singular point of {z E C : Q (z, z) = 0}.)
The Schwarz function is
=
b) COMPLEMENTS
OF UNBOUNDED
QUADRATURE DOMAINS
There are also unbounded domains Q satisfying identities of the kind (3.1 )
for all integrable analytic functions in Q ("unbounded quadrature domains").
There are even examples with p = 0 ("null quadrature domains"), e.g. halfplanes, exterior of ellipses and exterior of parabolas [Sa2], [Shl]. It can be
shown that (2.2) necessarily holds for an unbounded quadrature domain [Ah-Sh],
[Sal, Section 11 ], so the exponential kernel will vanish identically.
However, the complement of an unbounded quadrature domain often has a
nontrivial exponential kernel, which we now proceed to compute.
It is known [S111 ], [Sh2], [Sa4] that the class of those unbounded quadrature
domains which are not dense in the Riemann sphere coincides with the class
of images of bounded quadrature domains Q under M6bius transformations
with the pole on Q. It is enough to consider just inversions f (z) = 1 /z and
domains Q with 0 E Q.
So let Q be a bounded quadrature domain with 0 E S2 and data P, Q as in
the beginning of this section and let D
f (z) = 1 /z. Then (2.7), (3.6)
E nc)
give, for z, w E D (i.e., 1 /z,
_
=
For ~
we have
f(Q)c,
526
(cf. (3.8)), hence
Similarly,
Thus
we
(incorporatingP (0) ~ 2
find
into C)
for z, w E D.
EXAMPLE 3.3. The
ellipse.
Example 3.1
and f a Mbbius transformation with the pole
will be an ellipse, and all (genuine) ellipses can be
we get
obtained that way. Taking for example f (z) =
With Q
at the
origin
as
D
in
=
0 by (3.11) (S(,z) the Schwarz function of S2)
Since S(0)
together with (3.10) gives
=
EXAMPLE 3.4. The
equation (3.14)
parabola.
In similarity with the previous example, inversion of the quadrature domain S2 in Example 3.2 at the cusp point on
gives the exterior of a parabola.
we have
Indeed, with f (z)
I /z and D
=
=
527
The present example is a case in which formula (2.7) gives a nontrivial
result (i.e. C &#x3E; 0) despite the fact that the pole of the M6bius transformation
This is due to the fact that D is so small at infinity that ED =1= 0.
is on
Thus (3.14) applies again (with C &#x3E; 0) and gives, by (3.12), (3.13),
for Z, wED.
The results of Examples 3.3 and 3.4
D the interior of an ellipse or parabola
can
be
expressed by saying
that for
where q is a quadratic polynomial defining a D (D = f z E C : q (z, z) &#x3E;
0}), here normalized so that the constant C in (3.15), (3.16) disappears. The
formula (3.17) also holds for D an infinite strip, in which case q is a reducible
quadratic polynomial. For any other D with a D an algebraic curve of degree
two or less ED - 0 because of the size of D at infinity.
4. -
Analytic continuation properties and regularity of free boundaries
Let Q C C be a bounded domain and recall that the Cauchy transform
of Q is analytic in the exterior Qc. In this section we shall study the
has an analytic continuation
following type of question: Assume that
across aQ into Q, does then also
admit an analytic continuation
across aQ in one or both variables?
The answer to this question is in fact known to be "yes": By the work
of M. Sakai [Sa3], [Sa4] kQ extends analytically if and only if ao is analytic
with certain types of singularities allowed and for 8Q analytic the analytic
continuation of E has been proved by one of the authors [P3], at least when
no
singularities
are
present.
The classification by Sakai of boundaries 8Q admitting analytic continuation of X~ is quite long and technical. Here we shall reverse the order of
reasoning and first give a direct and relatively simple proof of the fact that
to say C B K, K C S2 compact, implies analytic
analytic continuation of
to ((~ B K)2. Then it follows from this fact that a SZ
continuation of
is analytic, more precisely that 8Q C {z E C B K : F(z, z)
0} where F is the
E.
of
continuation
analytic
In analogy with Section 3 we have that XQ I QC has an analytic continuation
across aQ if and only if S2 is a kind of "quadrature domain" ("quadrature
=
528
domain in the wide sense" in the terminology of [Shl], [Sh2]),
there exists a distribution it with compact support in Q such that
for every integrable
is equivalent to
analytic function (p
namely
in Q. Indeed, like in Section 3,
that
(4.1 )
A
is analytic outside supp /1.
In the case of classical quadrature domains (those in Section 3) there is a
beautiful argument of D. Aharonov-H.S. Shapiro [Ah-Sh] showing that, without
any apriori regularity assumption whatsoever, the boundary of such a domain
must be a subset of an algebraic curve. Once one has this global grip on a S2
the more detailed analysis of it (e.g. investigation of possible singular points)
is relatively easy.
One main point with our theorem below is that we do the same thing for
the general quadrature domains as Aharonov-Shapiro do for the classical ones:
we show that the boundary a S2 of such a domain must be contained in the
zero set of a certain real analytic function, namely the analytic continuation
of the exponential transform from outside. In the special case of classical
quadrature domains this function agrees (apart from some harmless one-variable
factors) with the Aharonov-Shapiro polynomial. Thus we feel that it is kind of
a "canonical" description of
that we obtain.
As a special case our result applies to many two-dimensional free boundary
problems in mathematical physics, e.g. the obstacle problem (with analytic data),
the dam problem, Hele-Shaw flow moving boundary problems etc. Indeed, in’
these problems one usually has more information about
to start with, namely
to
our
in
addition
to
our
(translated
context)
assumption (4.2) one has u = 0
on Ql, u &#x3E; 0 in Q B K where u is a certain real valued potential of XQ - ji
(so that au = XQ - p). See [Ro], [Sa5] for details.
Note, however, that our regularity result is not new. Sakai [Sa3], [Sa4]
starts from even weaker assumptions than we do. We essentially need a Schwarz
function S(z) satisfying the estimate (4.8) below whereas Sakai merely assumes
continuity of S(z). Also, we do not pursue the detailed analysis of possible
singularities of aS2 since this was done already in [Sa3], [Sa4].
and
a
THEOREM 4.1. Let S2 C C be a bounded open set and assume that there exists
C S2 and an analytic function f in K’ = C B K such that
compact subset K
Then there exists an
7~ x K~ such that
analytic-antianalytic function
F
(z, w) defined for (z, w)
E
529
Moreover,
More precisely,
setting A
F (z, z)
z
=
[Z
E
E
K :
A and F
SZ B
=
f (z) } we have
(z, z)
(4.3)
Like in Section 3, given the extension
tion [D], [Sh2] S(z) of aS2 to be
we
F
define the
(z, z) &#x3E; 0 for
A).
Schwarz func-
Then
Moreover, S is continuous.
satisfies
an
In fact, it is known that the
ifz- .z2 I e-2, hence the
where
d(z)
in the
sense
=
same
is true for
min(e-2, dist (z, Ql)). Finally
S(z).
In
particular,
note that
of distributions.
Next, following Ahlfors [Af] and Bers [Be]
function 1/1 E C °°R) satisfying 0 1/r~ 1, 1/1(t)
t
Cauchy-transform
estimate
&#x3E; 2 and define cut-off functions
choose a nondecreasing
1 for
0 for t 1, 1/1(t)
we
=
=
by
0 for z E Qc (n - 1, 2, ... ). Then each o/n is Lipschitz
Q, o/n (z)
continuous, has compact support in Q and 0
X~ pointwise as n -~ oo.
Moreover, we have a good bound on the gradient of pn :
for
z E
=
Combining (4.10)
with
(4.8) gives the important estimate
This enables us, e.g., to use a kind of Stokes’ formula without having any
a priori knowledge about the smoothness of aQ (besides the property (4.3)).
Indeed we have:
530
LEMMA 4.2. Lest 0 c C satisfy the hypotheses of Theorem 4. 1 and let 4) (z, w) be
is analytic and Lipschitz continuous in an open set in rc2 containing
a function
Then
E
{(Z,z) : Z
which
for any
open set D, K C D c
Q, with smooth boundary ( "smooth"
means
here
smooth
enough for Stokes’ formula).
NOTE: a D should be thought of as a freely moving contour just inside
then assuming smoothness of a D is of course harmless (as long as D
PROOF. By Lipschitz continuity and (4.11 ) we have
aQ and
c c
Q).
Thus, noting that a (S2 B D) n Q is smooth and equals a D n Q but has opposite
orientation, we obtain
By
the usual Stokes’ theorem
(since S(z) = z
on
applied
to the smooth domain D
Combining the two
from the proof, what
formulas
gives (4.12).
D
REMARK 4.3. As is clear
we really used about jo
is that there exists a continuous function S(z) defined and satisfying the estimate (4.8) in QBD, satisfying S(z) = z for z E aQ and being analytic in Q B D.
It is allowed in the lemma that part of a D agrees with aQ but then this
part of aQ must be smooth. Without that smoothness the result of the first
computation of the proof is still valid. This gives a more local version of the
lemma which applies e.g. to D = S2 B
r), Zo E a Q, r &#x3E; 0, whenever there
exists a Schwarz function as above in Q n
r).
531
The following crucial lemma is due to L. Caffarelli, L. Karp, A. Margulis
and H. Shahgholian. For completeness we include here most of the proof. The
main idea is found in [K-Mg]. It is based on an estimate by Caffarelli [Ca]
which has been generalized to our situation by Shahgholian (unpublished). The
last statement (b) is due to Karp (unpublished). The lemma generalizes in a
natural way to higher dimensions.
LEMMA 4.4 (Caffarelli-Karp-Margulis-Shahgholian).
ses in Theorem 4.1 then
If 0 satisfies the hypothe-
has area measure zero,
(a)
(b) Z = 0
(see (2.10) for the notation).
PROOF. By Lebesgue’s theorem, almost every point of aQ c Q’ is a density
point of Q’. Moreover, XQ is differentiable at almost every point [Cd-Z], [Re].
For the first statement of the lemma it is therefore
of points z E
at which both of
(i) QC has density one
(ii) XQ is differentiable
enough
to prove that the set
at z,
at z
hold has measure zero. We shall prove that this set is actually empty.
From this it will also follow that Z is empty, because (i) and (ii) do hold
for every z E Z. For (i) this was remarked after (2.11) and for (ii) we have,
using the same arguments as in the proof of Proposition 2.7 (the final estimates)
or directly invoking e.g. [St, p. 243-246], that
We thank L.
Now,
Karp and
satisfying (i)
and
(ii).
is differentiable at z
(Izl(
~
35(0)
=
the referee for useful comments in this context.
be a
(i) and (ii) are contradictory, let .z E
0. By (ii)
We may assume that z
to prove that
=
point
=
0, i.e.
0). Using (i) and that
f
on
S2~ it follows that
aS(O) = 0,
1, hence that
Next, fix any small
consider the functions
r &#x3E;
0, let
zn e Q
n D(0, r),
zn
-~
0 (n
~
oo) and
532
IS’(z)12 ~
for z
0, hence each un is
r). We have 88un (z)
monic. Since u, (z) = -lz-znI2
0 on
r), un (zn) = IS(zn) it follows that
0 somewhere on 8D(0, r) n Q.
Thus
=
n Q, and since zn
for some §n E
dicts (4.13). Hence the lemma is proved.
As
a
corollary
of
Proposition
2.7 and
-~
0,
I
-
b) of Lemma 4.4
PROOF OF THEOREM 4.1. We shall prove that there exists
function F (z, w) in Kc x Kc satisfying
E
(K’
x
K’) B Agz
and
0 this contra-
&#x3E;
D
COROLLARY 4.5. EQ is continuous everywhere and
when Q satisfies the hypotheses of Theorem 4. l.
for (z, w)
r
subhar-0
z)
an
we
=
have
0 for all Z
E
a Q
analytic-antianalytic
satisfying
for (z, w) E (KC x K’) B
In (4.14) it is understood that z-w
1 whenever w E QC, z-w
1
in
it
is
and
understood
in
whenever z E Qc (i.e., even if z
that,
w)
(4.15)
Q x Q, E(z, w)/Iz - wl2 is the analytic-antianalytic function H (z, w) appearing
in (2.15). Note that at each point of (K‘’)2 at least one of (4.14) and (4.15) is
in force. Indeed, both make plain sense, and say the same thing, in (X~)2 B A.
At each point of A KC exactly one of (4.14) and (4.15) makes sense as above.
Note also that (4.14) in particular says that F = E on
so F will indeed
be the required analytic-antianalytic extension of E. Since, by Corollary 4.5,
E
0 on Da also (4.5) will follow. The last statement of the theorem follows
by setting z = w in (4.15) and using (2.15), (2.18). Thus the theorem is proved
once F as above is constructed.
Next we remark that it is enough to prove (4.14), (4.15) for the set int Q
instead of Q.
Indeed, the difference set B = (int Q) B Q c a S2 has measure zero by
Lemma 4.4 so E does not recognize the difference (and neither does F). If
equations (4.14), (4.15) hold for int S2 they also hold for Q (and conversely)
because the change just means that the set of points AB is moved from the do0
main of validity of (4.15) to that of (4.14), and since S(z) = z and E(z, z)
which
is
valid
for
will
the
same
B
E
for z
AB
say
any equation (4.14), (4.15)
thing, namely that F = 0 on
-
=
=
=
_
=
533
Thus, renaming Q
to be int we may assume that
i.e., that each point of aS2 is accessible from nc (aS2
We will present two methods of constructing F satisfying (4.14), (4.15).
=
The first method is to write the integral in the definition of E as a boundary
integral, then perform the analytic continuation of E by deforming the contour
of integration and finally check that (4.14), (4.15) hold for the result F of the
analytic continuation.
The second method is to define F by (4.14), (4.15) and then check that F
is analytic-antianalytic.
METHOD 1.
We start
This is
a fixed w E -Qc and
a D smooth, such that
by taking
possible by continuity
of
S (~ ) and
choosing
an
open set D
c &#x3E;
=
0. It follows that
is single-valued analytic in
(we choose the natural branch of the
for which the argument is in the interval (-~/2, Tr/2)).
For any z E 52~, now apply Lemma 4.2 with
Since
this
gives
Dw with
logarithm,
534
As a function of z, the right member above is analytic in all D~. This means
that we have already produced a little bit of analytic continuation of E.
Next we want to continue the so obtained analytic function further, to C B K.
The integral in the last expression above defines one analytic function in D
and one other in C B D, say
Since a D is smooth the two functions he and h’ extend
and satisfy the jump condition
continuously
up to a D
for Z ~ aD.
Sinc’e
by
the
previous computation
it follows that the function
which is analytic in D B K, provides the further analytic extension of
w)
the path of integration a D
down to K. Finally one notices that, for
in the last integral can be moved back to
again. Thus the analytic extension
is
of
with
to
z
w)
given by
respect
F (z, W-) = (S(z) - W-) (z - w) exp[ -2.7riI- 1
I
for
z E
Q B K,
w E
log[(S(O - w) (’ - w)]
and where - aQ denotes any
path a D,
with
{z} U K
c
D cc Q.
_
Next we want to express F directly in terms of E. For
choose D(z, r) cc S2 B K. Since
w) = 1 (Example
4.2
Lemma
again
using
w) =
w)
2.3)
we
have,
535
Thus
forZEQ,
Next one observes that all the above works in the same way also if w E
It is just to replace Q by S2 B JI))(4V, r) (D(w, r) C C SZ B K) and use that
ED(w,r) (Z, w) = 1 for Z E D(w, r)c. The conclusion is that for fixed w E Q,
the analytic continuation of z H E(z, w ) from Q~ to (K U
is still given
the
member
of
(4.17).
by
right
the antianalytic continuation of w i--~
fixed E (I~ U
Similarly,
E(z, w) from ac to (I~ U fzl)c is given by
for
Combining the above two analytic continuations it follows that there exists
U (a Q x
which
analytic-antianalytic function F in (KC x K’) B
agrees with E in QC x QC and which satisfies (4.14), (4.15) in (Kc x Kc) B
x aS2)). Indeed, the joint analytic continuation from nc x Q~ to a
(A U
is achieved by first moving z to the
point (z, w) with say z V aS2 and
right place by (4.17) and then moving w according to (4.18). The latter means
that the expression (4.18) replaces E in (4.17) when w moves into Q.
Since the functions E(z,
wl2 and (S(z) - w)(z - S(w)) are analyticto the domain
antianalytic in all ( S2 B K ) x ( S2 B I~ ) (2.15) we can add
of validity of (4.15) by taking the right member of (4.15) as the definition of
F on
By this F is defined and analytic-antianalytic, and (4.14), (4.15)
x aQ).
hold, in (Kc x K~) B
The remaining continuation of F to all Kc x K~ is now automatic. Indeed,
an
let zo E
let r &#x3E; 0 denote the distance between 8Q and K and
wo E
let 0 s
1/2. Then for each Zl E D(zo, 8r) B aS2, wl E
Er) B aS2 the
has radius of convergence at least (1 - 8)r
power series of F (z, w ) at (zi ,
in each variable separately when the other variable is kept fixed.
By log-convexity of joint domain of convergence [Hm, Thm 2.4.3] the full
x
power series of F(z, w ) at (zi ,
converges in all D = D(zi ,
E
1/2 this domain contains (zo, wo). There
~/1problems
2013 ~r).of Since
multivaluedness for the analytic continuation because
will be no
the initial domain of analyticity D B
x
is connected and the extension
to D is defined by a single power series.
The above shows that F (z, w) has an analytic-antianalytic extension to all
K x KC. We must still check that (4.14_), (4.15) are valid on aQ x
i.e. that
F = E on
Q~
x
holds
in
F
E
both
and
are continuous
But this
x
and a Q x
c
by (4.16). Thus (4.14), (4.15) hold as stated and
by this the construction of F by Method 1 as finished.
536
METHOD 2.
Define F(z, w) at each point (z, w) E K~ x KC by (4.14), (4.15). We shall
prove that F is analytic-antianalytic. For this, in the virtue of Hartog’s theorem [Hm, Thm 2.2.8], it is enough to prove that F is analytic (antianalytic) in
z (resp. w) when the other variable is held fixed and by symmetry it is enough
to consider one of the cases, say to prove analyticity in z when w is held fixed.
So let w E K~ be fixed. The factor
is clearly analytic in z, except possibly for a simple pole at z
w in the case
that w E Q. Therefore it is enough to show that the function G(z)
Gw (z)
defined in KC by
=
=
is analytic and, in case W E Q, has a zero at z
w. As to the interpretation
of (4.19) on the diagonal similar rules as for (4.14), (4.15) apply: if z = W E QC
the first expression shall be used, saying that G
E then, if z = w E Q the
second expression shall be used.
The analyticity of G in nc and in ~2BA~ and that it vanishes at w if W E Q,
is immediately clear from the general structure (2.15) of E (z, w). Problems
can arise only on
Since G is locally bounded except for the factor 1 / (z - w ), it is locally
integrable, in particular G is a distribution in Kc. We shall compute aG in the
sense of distributions. Let { ~n { be the Ahlfors-Bers cut-off functions (4.9). By
G E
S(z) - z 0 on Qc we have, as n - oo,
=
=
Lloc,
in the
=
of distributions (use e.g. dominated convergence
action on test functions). It follows that
sense
expressions
on
the above
in the sense of distributions.
Due to the factor 1/In (z) everything in the second term in (4.20) takes place
inside a compact subset of Q, where S(z) - z is smooth and E(z, w)/ (z - w)
is analytic in z by (2.15). Therefore we are allowed to use the product rule for
the derivative, giving that
.
537
In view of the estimate
(4.11 ) and the integrability of E (z, w) / (Z- - i-v) the first
(in the sense of distributions) as n ~ oo and the
w)/ (z - w). Taking the first term in (4.20)
and using (2.19) we therefore conclude from (4.20) that
term above tends to zero
second term tends
into account
aG
=
0.
Thus G(z) is analytic in K’ as a distribution, i.e. there exists an analytic
function in K’ which differs from G at most on a nullset. The final observation
now is that this nullset has to be empty. Indeed, we claim that if we change
the definition of G on any nonempty nullset the result will be a discontinuous
function.
int S2 and the fact that E is
For this we have to use the assumption S2
continuous everywhere. By definition (4.19) G is continuous in S2 and equals
E(’, w) on QC. By the above remarks the restriction of G to QC therefore is
continuous and each point of QC is in the closure of the open subset
It
follows that if we change the definition of G at any point zo,
or
in Qc, but leave it unchanged almost everywhere it will become discontinuous
_
=
at zo.
Thus
tion,
conclude that G already from the beginning was analytic as a funconly as a distribution. This finishes the Method 2 proof of Theorem 1. D
we
not
REMARK 4.6. By small modifications of the proof one arrives at a local version
of Theorem 4.1 saying the following: if Q, U c C are open (think of U as
a neighbourhood of some point on a Q) and there exists f analytic in U such
that XQ = f in U B Q then there exists F (z, w) analytic-antianalytic in U x U
F in (U B Q) 2. Moreover, aS2 n U c {z E U : F (z, z) = 01.
such that E
=
REMAK 4.7. The role of part a) of Lemma 4.4 in the proof of Theorem 4.1
is that it permits us to work with int S2 instead of Q. Part b) of the lemma
is really not needed, it is enough to use that a S2 B Z is dense in
(see
before Proposition 2.7). If one is willing to assume in advance that Q
int S2
then Lemma 4.4 can therefore be dispensed with and the proof of Theorem 4.1
becomes shorter. Moreover, assumption (4.3) may in this case be relaxed to
=
because
5. -
(4.3) will follow from (4.21) by continuity.
Analytic continuation of resolvents of hyponormal operators
be the unique irreducible hyponormal operator with rankLet
self-commutator (i.e., [T*, T]] = 0 for some nonzero E H) and
with principal function equal to the characteristic function xgz of a bounded
one
538
domain S2. Here H is
°
a
separable complex
Hilbert space with inner
product
denoted (.,.).
It is well known that a generalized weakly continuous resolvent (T*
exists for every z E C, see [MP, Ch. XI]. Moreover, a fundamental
established by K. Clancey [Cll] asserts that
- z)-l
~
identity
Starting with the same assumption as in Section 4 (namely that the Cauchy
transform XQ extends analytically across the boundary of S2) we will prove that
the resolvent (T* - Z-)-l~ itself extends antianalytically across
This will
in
4
of
the
main
result
Section
a
little
bit
more.
In the
provide another proof
and
int
S2
=
Q
section
we
will
for
assume
that
Remark
(cf.
4.7).
present
simplicity
Assume that
f (z), z g Q, where f is analytic in C B K and define
as above the corresponding Schwarz function
=
Let
and
1/In be the Ahlfors-Bers exhaustion functions (4.9) satisfying 1/In /
(4.11 ).
LEMMA
5.1. The following limit exists in
PROOF. Let w
By passing
Recall that
to
E
D ((C) and let n
the limit
we
E
1)’ (C)0H:
N be fixed. Then
find
1,
Z ~
C,
see
[MP].
XQ
539
We consider the closed linear span
and the restriction R *
R EC(HI) and
In
particular a (R) C S2
onto
=
=
T * ~Hl (of T *
a(T). Let
to this invariant
P denote the
subspace).
orthogonal projection
Then
of H
Hl.
PROPOSITION 5.2. The distribution
is
antianalytic in C B
PROOF. Let
w
K.
g SZ,
and consider the scalar valued distribution
Gw (z) defined by (4.19). But according to Section 4, Gw (,z) is analytic for
satisfies
z fj K. Hence the distribution
Since the space HI is
for
generated by
the vectors
(T* -
w)-l~,
w / Q, it follows
0
540
COROLLARY 5.3. The Cauchy transform is2 (z) extends analytically fr_om QC to KC
if and only if the resolvent (T* - z) -1 g extends antianalytically from QC to Kc.
PROOF. One
is
antianalytic,
implication
follows from
and it satisfies
Proposition
5.2. Indeed, the function
for z ¢ S2
whence V(z) is an antianalytic extension to KC of the resolvent (T* - z)-1~.
Conversely, suppose that the antianalytic extension V (z) of the resolvent
(T * - Z-) - 1 ~ exists for z ¢ K. Then, for z ¢ Q, it is known as a consequence
of Helton-Howe trace formula, see [MP], that
whence the
analytic
extension of
is
given by
the function
(~, V(z)).
0
THEOREM 5.4.
Assume that the Cauchy transform of the bounded domain S2 anaw)
lytically extends from S2~ to Kc (K C C Q). Then the exponential kernel
extends
analytically from S2~
S2~
x
to
KC
x
KC and its extension is
given by the
formula
V is the function defined in Proposition 5.2).
Moreover, F (z, z) &#x3E; 0 for z fj Q and F (z, z) 0
(where
z
PROOF. Only the last statement needs
fj K, it follows that
a
proof.
for z
Since
E
SZ B
K.
(T* - z) V (z) = g for
x is by definition the vector of minimal norm satisfying
because (T* - z)-1 ~
1
In addition, we know that"( T * - z ) -1 ~ ~ ~I
the equation (T* - z)
1 for z E Q. Moreover, V(z)
for z ¢ S2
(T * - z)-1 ~,
0
z V Q, and the statement follows.
=
x-~.
=
=
COROLLARY 5.5. Let F (z, W-) be the analytic extension of the exponential kernel
w). Then the function 1 F (z, w- ) is positive semidefinite.
-
541
PROPOSITION 5.6.
PROOF. Since
for every
The
model
A
0
pair (z, w) E Q x Q, the estimate follows from (2.9), (2.14).
following semidefinite scalar product on D(C) appears in a functional
which diagonalizes the operator T*, see [MP]
simple application
of Stokes’ theorem leads to the
following
result.
LEMMA 5.7. Let S2 c C~ be a bounded domain with piecewise smooth
and let cp, 1/1 E D(C), Xn E D(S2), Xn / XQ. Then
We omit the details.
on
the
completion
of
We
finally
boundary
remark that the operator T is rationally cyclic
to the above scalar product, see [P2].
D(C) with respect
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Department of Mathematics
Royal Institute of Technology
S-100 44 Stockholm, Sweden
gbjornC math.kth.se
Department of Mathematics
University of California
Santa Barbara, CA 93106, USA
[email protected]
.